Slide rule



Mx M. w. 1

SLIDE RULE mal Filed oct. v1'7, 1944 J. R. BLAND Orig Patented Dec. 19, 1950 UNITED STAT-ES orrlce SLIDE RULE J ames R. Bland, llastport Md., assigner to Konf- =fel -& EsserfCompany, Hoboken, N. J., a corporation of New Jersey (Cl. 235--7m IClaiin. 1

This invention relates to'slide rules and vhas as its `object to provide a rule Aof enhanced power Yand in the use of which .by a process requiring ygenerally one but no more than one .movement :either of vthe hairline .or of the slide for each .number lin the expression to be computed. The improvedcapabilities of the newrule can -be best .-explaned with reference to .an actual embodiment such as is .shown by -way of illustration in the accompanyingdrawing in `which:

Figure l is a 4view ofone face of :a slide rule in .accordance with the invention and Figure2 isa-V'eW of theother face.

As a matter of :convenient reference, the face of therule shown -in .Figure l will hereinaiterbe `considered to `.be the `front face .and the face shown .in .Figure .2 to be .the rea-r face. I have chosen to show the inventionasapplied to a rule which is the samein mechanicalstructureias .that shown in Patent No. 2,170,144, grantedvAugust 22, 1939. -As in the said patent, reference .letters C.scale, `graduated `in .accordance .with the logaf .rithms ,of numbers .from .1 .to 10, and .of unit length. The bodycarries 1the usual Dfscale which is exactly the same as the C scale .and .these .two scales .are herein considered .to be basic Vscales since all other scales are directly associated .with I them so that .all slide rule operations are most easly explainahle Yby .means .of them.

Above the .Cscale -as viewed-inFigure Lisa CI scale which is 'identicaland co-.extensive `with the C scale except that it is inverted with respect thereto. 'The term co-extensive as used herein includes the meaning of registry, i. e., the inclusion Within common parallel terminal lines, 'insofar as eiiective extent is concerned.

Above the CI scale is the CIF scale which is the same as the CI scale except that it is folded at 1r, while the immediately adjacent CF scale is identical with the `C :scale except that it is folded at 1r. The immediately adjacent DF scale, on the body, is identical Iwith ,the :CF scale.

On its rear face, the slide carries a B scale which is co-extensive with the C scale and is graduated in accordance with the 'logarithme oi numbers from 1 to 100. Below the B scale, as Viewed in Figure v2, the slide carries xthe rco-extenyspends to the range used furthe LL scales.

sive .trigonometric scales T, ST and S. `The T scale is graduated vin accordance with the logarithms c'f the values ofthe natural tangents from 5 a3 to 45, theST scale is graduated in accordance with the logarithms of the values of the natural lsines of the angles from 34 to 5 43', while the S scale is like the ST scale except that its range is from 5 43' to.90.

On its rrear face, the bodycarries K, A, D and 'L scales all Ico-extensive with the first-mentioned entire scale is graduated in accordance with `the l'ogarithms of the logarithms of numbers greater than unity. Finally, the rule Ais provided with an LLO scale in three sections each of full unit length oi which section LLOl appears on vthe rear 4face and :sections LLO2 and YLLOS appear on the front face. The sections'o'this scale are each (zo-extensive with the basic scales and the scale as a whole is Agraduated in accordance with the logarithms of the yco-logarithms of the posi- -tive numbers :less than unity, or, to express it differently, in accordance with the logarithme of the logarithms of the recprocals of thenumbers :of the LL scale.

The `range selected 'ior the LLO scales corre- In the Aillustrated embodiment e and e-'l are opposite an index of the D scale. In the `case of a rule yin which lit is necessary to conserve space'on the face of the rule, la different range of Values could be selected for the LL scales and a corren spending range would be selected for the LLO scales. In such a case, some number other than e would be placed opposite vthe index of D, front race. The choice in this matter would depend upon a selection ci the most useful range-of values on the LL anden the LLO scales. For example, if a single LL scale section were to be used,`the range of values chosen might be from 1.585 (approximately) to 196. ln this case, the lirst of these numbers would be set opposite the left index of D, while lthe 1GO would be set opposite the right index of D, With this choice for a Asingle 'line LL scale, the co-extensive single line LLO scale would have the range of 0.631 (approximately) to 0.01. The first of these numbers would be placed opposite the left index of D, while the point 0.01 would be set opposite the right index o" D.

As here shown, the LLO scale is graduated in accordance with the logarithms of the co-logarithms of positive numbers between e-U-01 equals 0.9900, and e-1 equals 0.9048, e-1 and e-1 equals 0.3679 and e*1 and e10 equals 0.000052, respectively. These graduations are so spaced that, like the LL scale, the trigonometric scales and the K, A, B, L, DF, CF, CIF and CI scales, the LLO scale is directly associated with numbers on the basic scales C and D. By directly associated, we mean that when the rule is closed (i. e. when the indices are aligned) and the hairline is pushed to a value on any scale other than the C and D scales, the value or the function of this scale associated with the marked value is directly readable at the hairline on scale C, For example, in the case of the trigonometric scales, if the hairline is pushed to an angle on the S scale, the sine of that angle will be marked at the hairline on the C scale; if the hairline is pushed to a number on the B scale, the square root of the number will be marked at the hairline on the C scale; if the hairline is pushed to a number on the CI scale, the reciprocal of that number will be marked at the hairline on rthe C scale; and if the hairline is pushed to a number on the CF scale, the hairline will mark on the C scale the number divided by 1r.

A great advantage arises from the direct association of the LLO scale with the basic scales C and D instead of with the scales A and B, which latter has been the practice heretofore as illustrated, for example, in the above mentioned patent. Every scale outside of the C and D scales is directly associated with the C and D scales. Consequently, the rules governing the placing of the decimal point and the result arrived at in the use of the LLO scale are very similar to those governing the placing of the decimal point in a result arrived at in the use of the LL scales. Operating rules of a very unlike and therefore confusing nature govern the use of the two corresponding sets of Log Log scales in prior art rules of the type shown in the said patent.

The new LLO scale gives additional power to the rule and makes the operator more accurate in making settings, since the principles are similar to those involved in other settings. and this simplicity is achieved by the described association of the LLO scale with the basic C and D scales. The operator has the advantage of being able to obtain negative powers of numbers greater than 1, and negative powers of numbers less than 1 simultaneously. The LL and LLO scales may be used in cooperation with the C and D scales in one operation. For example, if the operator wishes to i'lnd 1*2, he sets the left index of C opposite 4 on the LL3 scale, pushes the hairline to 2 on the C scale, and at the hairline reads 0.0 62 on the LLO3 section.

The advantages arising from the new association of scales will be evident from the following examples:

Eample 1.--Evaluate: 3.470-1, 3.47-0-1, 3.47001, and 3.47--01.

Solution-Set hairline on glass indicator to 3.47 on scale LLB At hairline on LL2 read 3.47-1=1.1325 At hairline on LLO2 read 3.47-0-1=0.883.

4 At hairline on LL1 read 3.47-1=1.0125 At hairline on LLOl read 3.4701=0.9877

Example 2.-Find the logarithms to the base e (=2.7183 approximately) of the numbers 1.135, and its reciprocal 0.883, 1.0125, and its reciprocal 0.9877.

Solution- Set hairline on glass indicator to 1.135 of scale LL2 At hairline 0n D scale read loge 1.1325 (found on At hairline on D scale read loge 0.883 (found on LLO2) =-0.1242

At hairline on D scale read loge 1.0125 (found on At hairline on D scale read loge 0.98765 (found on LLO1)=-0.01242 Observe in Example 1, that the answers for like exponents were found on like numbered scales, and in Example 2 that the decimal point in the answers were similarly placed when the answers were derived from like numbered scales. In prior art rules, no such simplicity obtained because the scales marked LLOO and LLO were read against the A and B scales whereas the scales marked LL1, LL2, and LL3 were read against the C and D scales for practically every important problern. It was a matter of common experience, that students had great difculty in using the scales marked LLOO and LLO on prior art rules because of difficulties arising because the A and B scales were repeated scales. These diculties do not arise in the operation of the slide rule of the present invention, since like rules of operation apply to both sets of the Log Log scales.

In the use of the new slide rule most practical applications of the Log Log scales are operated in conjunction with the C and D scales. However many important applications involve their use with other scales. In every case likeness of method of operation and of placing the decimal point obtain when the rule of this invention is employed. The following examples will illustrate this fact.

Example 3.-Evaluate esin 3 and e-S 30 Solution- Using the rule of the present invention, close the rule, that is, set the left mark numbered 1 on scale C opposite the same numbered left mark of scale D, push the hairline on the glass indicator to 30 on scale S, at the hairline on scale LL2 read 1.65 and on scale LLO2 read eS1n,30=0.606. Observe that like numbered Log Log scales LL2 and LLO2 were used.

Example 4.-Evaluate 5cos "0 and 5- C05 '10".

Solutz'0n.-Opposite 5 on scale LLB draw left index of C, push hairline to 70 (red) on scale S, at the hairline on LL2 read 1.734=5c"s '10 and at the hairline read 5*cos 7=0.577 on LLO2.

Example 5.--Evaluate 0.2tan 25 and 0.2-11 25.

Solution- Opposite 0.2 on scale LLO3, set 1 of scale C, push hairline to 25 on scale T,

At hairline read 0.2tml 25=0A72 on scale LLO2 At hairline read (l2-ifm 25- 2.12 on scale LL2.

Example 6.-Evaluate o.155-2 and 015i2 Solution- Draw index of CI opposite 0.15 scale LLOS. push hairline to 5.2 on CI,

At hairline read 0.155'2=0.694 on Scale LLO2 At hairline read (0.15) 5'2=1.44 on scale LL2 e525-eases Example 7..-.-Eva'luate 08m/ganci 0;80\/ Solution-Draw left' index of B opposite 0.80 on scale LLO2, push hairline to- 5 on leithalr of B,

At hairline read o.sc\/ =o.6o7 .on scale LLoz, At hairline read 0.-80-\/5=11.647 on LL2.

Example 8.--Evaluate 343-3 and c3455 Solution-Push the hairline tor 33 on the vmiddle K scale,

A@ hairiine read faire: .040.4 on LLos.

At hairline read @Si/55:24.? on LL3.

Solution-Push the hairline to .624 lon scale L At hairline readV crantilog106-24=0-656 on scale LLO2 At hairline read emi1g106-24=1-524 on scale LL2.

Example 10.--Evaluate 2.16/n and 2.1-W1r Solution-Draw right index of CF to 2.1 oi

LL2, push the hairline to 6 of CF At hairline read 2.16/=4.12 on LL3 At hairline read 2.1-6/"=0.242 on LLOB.

Example 11.-Evaluate L (0.25)27r and, (0.25)-0-2" Solutioa.-Draw index of CIF opposite .25 of LLOB push hairline to 2 of CIF 1 in hairline read o.25)2=o.11 on LLos l A@ hairline read (c.25)02"=9.1 on LL3 The examples given above show the same likeness of operation and of placing the decimal point for the Log Log scales in conjunction with each of the other scales on the rule. Examples for the A, D, DF, and ST scales are not given since they would duplicate in principal examples already given.

The slide rule of the present invention has more power than the prior rules as regards the Log Log scales. For example, the expression 0.2*tim 25 occurring in Example 5 could not be evaluated with a single setting of the slide on the prior art slide rules, neither could such expressions as 2.903 and 0.29-0-31. These expressions involve the use of scales LLl, LL2 and LL3 in conjunction with the C scale and the scales LLOl, LLO2, and LLO3 on the rule of this invention and are easily evaluatedwith a single setting of the slide. rFhus to evaluate 2.9-0-3 Draw the index of C opposite 2.9 on LL3, Opposite 3 on C read 2.93=0.726 on LLOZ and to evaluate 0.29-0-31 Draw the index ofC opposite 0.29 on LLGS Opposite 31 on C read 0.2931=1.467 cn LLZ.

Hyperbolic functions are being used more and more in engineering practice. With the new LLOl, LLO2, and LLO3 scales in conjunction with the scales marked LLl, LL2, and LL3, it is a simple matter to find the value of various hy perbolic functions. Thus to find sinh 2 and cosh 2, push the hairline to 2 on scale D and read at the hairline on LL3 7.39 and on LLOS 0.135. Hence sinh 2=1/2(7.390.135)=3.627 and cosh 2=1/(7.39}0.135)=3.762. It is to be observed in this process that sinh;2 and cosh 2 were easily obtained since both the numbers 7.39 and 0.135 were found at once on Log Log scales LL3 and LLOB having like numbers when the hairlinewas pushed to 2 onscale D. With the prior art slide rules, the number 7.39 -is found as above and the number 0.135 is found opposite 2 of one of the left A scales on the LLO scale. .Instead of a single setting of the hairline on the rule of the present invention, two settings are necessary on the prior art rules and complicated rules of determining what scales apply are involved.

As illustrating the obtaining of a final result by continuous progressive"manipulations of the new slide rule in the evaluation of expressions involving logarithms of numbers less than unity, which evaluation would require resetting .in the use of any prior art slide rule, the following examples are given:

Example 12.-Evaluate 17 log.. 0.04

Schiuma-To 0.01 on LLOB scale Draw 31 on B (right) scale, Push indicator' to 17 on CF scale and Read 9.83 on DF scale.

Example 13.-Evaluate Draw 70 on S (red), Push indicator to 45 on iS scale and Read .6955 on D scale Example 14.-Eva1uate cos 79 05 loge 0.9525 sec 70 10 f sin 55 Solution-To 0.9525 on LLOl Draw 55 on S, Push hairline to 79 05 on S (red), Draw 70 10' on S (red) to hairline and Read at the index .03319 on D.

Example 15.--Evaluate \/16.2 loge 0.0074 csc 60 Solution-To 0.0074 on LLOB scale Draw 60 on S scale, Push hairline to 16.2 on B and Read 22.8 on D scale.

Observe that it is impossible, by means of prior art rules of any kind, to evaluate each of the expressions in Examples 12, 13, 14. and 15 by continuous progressive manipulations, that is by a process requiring generally one but no more than one movement either of the hairline or of the slide foreach number in the expression to be computed.

Example 16.-Evaluate '-log, 0.72 tan 25 e sin 40 ks-5 Solution-Push hairline to 0.72 on scale LLO2,

Draw 40 of S scale under the hairline,

Push the hairline to 25 of T scale,

Draw 35 of B (right) under the hairline,

Push the hairline to index of C,

At the hairline read 1.0412 on LLl scale.

Example 17.-Evaluate Solution-Push the hairline to 1.311 on scale LL2,

Draw 47 of scale S (red) under the hairline,

Push the hairline to 40 on scale T,

Draw 3.7 of scale CIF under the hairline,

Push the hairline to 35 of scale T,

At the hairline on scale LLO3 read 0.0664.

Ercample 18.-Eva1uate (log. 0.65)(1/) e sin 40 Solution- Push hairline to 0.65 on scale LLO2,

Draw 40 of S scale under the hairline, Push the hairline to 23 on B right, At the hairline read 0.0401 on scale LLO3.

Two resettings would be necessary to perform this evaluation by means of the slide rule of Patent No. 2,170,144.

Another feature of improvement over the above mentioned prior patent lies in the disposition ol? the sine, co-sine, tangent and co-tangent numbers. In said prior patent, the co-sinc and cotangent numbers are in red and are to the right vof the markings, whereas the sine and tangent mon marking. The transposition of the associated numbers and their upward divergence greatly facilitate accurate settings.

It is Well understood that in slide rules the slide may be rectilinearly movable in a slot, as herein shown, or in a channel, or the rule may be in disc or cylinder form with a rotary slide member. When in the following claims a slide member is recited, it is to be understood that it may be of any of the known types. Variations from the specific disclosure herein, for example in the relative positioning of scales, are possible and are contemplated in the claims which follow.

This is a division of application Serial No. 559,020 Jiled October 17, 1944, entitled Slide Rule, now Patent 2,422,649, granted June 17, 1947.

What is claimed is:

In a slide rule, a trigonometric scale having graduations designated on the right by a series of numbers ascending from left to right and on the left by a complementary series of numbers descending from left to right, the numbers associated with each graduation diverging upwardly with respect thereto. 4

JAMES R. BLAIID.

REFERENCES CITED U'NTED STATES PATENTS Name Date Bernegau Aug. 1, 1939 Number 

